A318175 Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function.

1, 2, 8, 9, 10, 15, 18, 21, 24, 30, 42, 60, 144, 160, 168, 240, 270, 288, 324, 480, 512, 630, 648, 960, 1023, 1200, 1404, 1428, 1536, 2046, 2400, 2808, 2856, 2880, 3276, 3570, 4092, 4320, 4608, 6552, 8925, 10080, 10368, 10752, 11550, 13824, 14280, 14976, 15345, 16368, 17850

Offset

1,2

Comments

As in A019278, here there are many instances where if x is a term, then A188999(x) is also a term.

Additionally, there exist longer chains of 3 or 4 elements; e.g.,

- 8 (3), 15 (4), 24 (5), 60 (6);

- 9 (2), 10 (3), 18 (4), 30 (5);

- 512 (3), 1023 (4), 1536 (5), 4092 (6);

- 8925 (4), 14976 (5), 35700 (6);

- 219969739395000 (16), 899826278400000 (17), 3519515830320000 (18).

Links

Giovanni Resta, Table of n, a(n) for n = 1..227 (terms < 10^12; first 185 terms from Tomohiro Yamada)

Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017. See Table 1.

Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.

Michel Marcus, Unexhaustive list of terms.

Example

For m=2, A188999(2) = 3 and A188999(3) = 4, so 2 is a term with k=2.
For m=9, A188999(9) = 10 and A188999(10) = 18, so 9 is a term with k=2.

Mathematica

bsigma[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]];
Reap[For[m = 1, m < 20000, m++, If[Divisible[bsigma @ bsigma @ m, m], Sow[m]]]][[2, 1]] (* Jean-François Alcover, Sep 22 2018 *)

Prog

(PARI) a188999(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = frac(a188999(a188999(n))/n) == 0;

Crossrefs

Cf. A188999 (bi-unitary sigma).

Cf. A019278 (analog for sigma), A318182 (analog for infinitary sigma).

Sequence in context: A167450 A050569 A352698 * A318182 A047469 A283774

Adjacent sequences: A318172 A318173 A318174 * A318176 A318177 A318178

Keyword

nonn

Author

Michel Marcus, Aug 20 2018

Status

approved